Pointwise error estimates for Legendre spectral differentiation of functions with limited regularity in fractional spaces

Legendre spectral differentiation is a basic tool in the numerical solution of differential equations. More precise information on spectral differentiation errors is important for deriving reliable error estimates for related numerical algorithms. However, existing studies primarily focus on asymptotic error estimates for polynomial approximations of specific singular functions and lack sharp pointwise error estimates for functions with interior or endpoint singularities in fractional spaces. In this work, we present explicit and sharp pointwise error estimates for Legendre spectral differentiation of functions with limited regularity in fractional spaces. We start by specifying a fractional-space setting suited to the pointwise error analysis in order to deal with the pointwise error estimates. We then derive explicit upper error bounds for Legendre spectral differentiation of functions with interior or endpoint singularities. Numerical experiments are provided to demonstrate the sharpness of our results.